Computing Jacobi's $\theta$ in quasi-linear time

Jacobi’s $\theta$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $\theta (z,\tau )$, for $z,\tau$ verifying certain conditions, with precision $P$ in $O(\mathcal{M}(P)\sqrt{P})$ bit operations, where $\mathcal{M}(P)$ denotes the number of operations needed to multiply two complex $P$-bit numbers. We generalize an algorithm which computes specific values of the $\theta$ function (the $\mathit{\text{theta-constants}}$) in asymptotically faster time; this gives us an algorithm to compute $\theta (z,\tau )$ with precision $P$ in $O($\mathcal{M}$(P) \log P)$bitoperations,forany$\tau \in $\mathcal{F}$$and$z$ reduced using the quasi-periodicity of $\theta$.